Chapter 2

Algebra and Notation

The reader should be familiar with most of the topics in this chapter. However, it is oftenthe case that set notation is not familiar and so a short discussion of this is included first.Complex numbers are then considered in somewhat more detail. Many of the applicationsof linear algebra and differential equations require the use of complex numbers, so this isthe reason for this introduction.

2.1 Sets And Set NotationA set is just a collection of things called elements. Often these are also referred to as pointsin calculus. For example {1,2,3,8} would be a set consisting of the elements 1,2,3, and8. To indicate that 3 is an element of {1,2,3,8} , it is customary to write 3 ∈ {1,2,3,8} .9 /∈ {1,2,3,8} means 9 is not an element of {1,2,3,8} . Sometimes a rule specifies a set.For example you could specify a set as all integers larger than 2. This would be written asS = {x ∈ Z : x > 2} . This notation says: the set of all integers, x, such that x > 2.

If A and B are sets with the property that every element of A is an element of B, thenA is a subset of B. For example, {1,2,3,8} is a subset of {1,2,3,4,5,8} , in symbols,{1,2,3,8} ⊆ {1,2,3,4,5,8} . It is sometimes said that “A is contained in B” or even “Bcontains A”. The same statement about the two sets may also be written as {1,2,3,4,5,8}⊇{1,2,3,8}.

The union of two sets is the set consisting of everything which is an element of at leastone of the sets, A or B. As an example of the union of two sets {1,2,3,8}∪{3,4,7,8} ={1,2,3,4,7,8} because these numbers are those which are in at least one of the two sets.In general

A∪B≡ {x : x ∈ A or x ∈ B} .

Be sure you understand that something which is in both A and B is in the union. It is not anexclusive or.

The intersection of two sets, A and B consists of everything which is in both of the sets.Thus {1,2,3,8}∩{3,4,7,8}= {3,8} because 3 and 8 are those elements the two sets havein common. In general,

A∩B≡ {x : x ∈ A and x ∈ B} .

The symbol [a,b] where a and b are real numbers, denotes the set of real numbers x,such that a ≤ x ≤ b and [a,b) denotes the set of real numbers such that a ≤ x < b. (a,b)

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